You have the section of the smaller circle wrapping around the back of a narrower section of the cone, a right angle and then a straight line running down the side of the cone, another right angle in the other direction, then the section of the larger circle, this time going around the front of the cone, another right angle, and then the straight line completing the shape.

But you are right, the circle sections are not geodesics on that manifold, so not ‘straight’ by the most reasonable extension of that word to non -euclidian geometries. They’d be more like lines of latitude in that they seem straight from the outside, but in the context of that manifold are curved.